Fermionic criticality of anisotropic nodal point semimetals away from the upper critical dimension: exact $1/N_f$ exponents

TL;DR

This paper investigates the quantum critical behavior of anisotropic nodal point semimetals with strong interactions, deriving exact critical exponents in certain cases and highlighting the importance of proper cutoff schemes in RG analysis.

Contribution

It introduces a soft cutoff RG approach to accurately determine universal critical exponents for anisotropic fermionic systems, correcting previous non-universal results.

Findings

Exact critical exponents for semi-Dirac fermions computed to leading order in 1/N_f.

Reproduction of known exponents for relativistic Dirac fermions using the new method.

Demonstration that cutoff scheme choice significantly affects RG results in fermionic quantum criticality.

Abstract

We consider the fermionic quantum criticality of anisotropic nodal point semimetals in $d = d_L + d_Q$ spatial dimensions that disperse linearly in $d_L$ dimensions, and quadratically in the remaining $d_Q$ dimensions. When subject to strong interactions, these systems are susceptible to semimetal-insulator transitions concurrent with spontaneous symmetry breaking. Such quantum critical points are described by effective field theories of anisotropic nodal fermions coupled to dynamical order parameter fields. We analyze the universal scaling in the physically relevant spatial dimensions, generalizing to a large number $N_f$ of fermion flavors for analytic control. Landau damping by gapless fermionic excitations gives rise to non-analytic self-energy corrections to the bosonic order-parameter propagator that dominate the long-wavelength behavior. We show that perturbative momentum shell RG leads to non-universal, cutoff dependent results, as it does not correctly account for this non-analytic structure. In turn, using a completely general soft cutoff formulation, we demonstrate that the correct IR scaling of the dressed bosonic propagator can be deduced by enforcing that results are independent of the cutoff scheme. Using this soft cutoff approach, we compute the exact critical exponents for anisotropic semi-Dirac fermions $(d_L=1$, $d_Q=1)$ to leading order in $1/N_f$, and to all loop orders. Applying the same method to relativistic Dirac fermions, we reproduce the critical exponents obtained by other methods, such as conformal bootstrap...